Page 118 - Textos de Matemática Vol. 44

108 X. LIU AND T.-Y. TAM
A∈Cn×n forc∈Cn definedby n
Wc(A) = cix∗i Axi : x1,...,xn are orthonormal in Cn . i=1
Westwick proved that Wc(A) is always convex if c ∈ Rn and fails to be convex if c ∈ Cn in general. Let c = (c1,...,cn)⊤ and C = diag(c1,...,cn). Then one sees that μ ∈ Wc(A) if and only if μ ∈ tr(CU∗AU) for some U ∈ U(n), where U(n) denotes the unitary group. This observation motivates the definition of C-numerical range of A ∈ Cn×n for a general C ∈ Cn×n defined by
WC(A) = {tr(CU∗AU) : U ∈ U(n)}.
This notion was first introduced by Goldberg and Straus in [11]. Note that
WC (A) is the image of the unitary orbit
U(A) := {U∗AU : U ∈ U(n)}
under the linear functional on Cn×n represented by C. Clearly WC(A) = WA(C) and WC(A) = WC(U∗AU) for all U ∈ U(n). Cheung and Tsing [7] proved that WC(A) is star-shaped.
LetC∈Cn×n beHermitianandletA∈Cn×n.LetA=A1+iA2 bethe Hermitian decomposition, where A1 and A2 are Hermitian. Then WC(A) can be identified with
WC(A1,A2) := {(trCU∗A1U,trCU∗A2U) : U ∈ U(n)} ⊂ R2.
Note that U(n) is a compact connected Lie group, whose Lie algebra u(n) consists of all n × n skew Hermitian matrices. If B ∈ Cn×n is Hermitian, then iB, iC ∈ u(n) and
tr(CU∗BU) = tr(BUCU∗) = −tr(iB)U(iC)U∗.
Thus one can assume that A1,A2,C ∈ u(n) when concerning convexity of WC (A1, A2).
Westwick’s proof uses the idea of Hausdorff’s connectedness argument. He
considered the function fB,C :U(n)/D(n) → R given by fB,C([U])=trCU∗BU,
where B, C ∈ Cn×n are Hermitian, D(n) is the subgroup of diagonal matrices
in U(n), and [U] = D(n)U for U ∈ U(n). He showed that f−1 (c) is connected B,C
for any c ∈ R. His main idea is to show that when B and C have distinct eigenvalues, fB,C is a Morse function and its Hessian has even index at each critical point. Then handle body decomposition is used to finish the proof. However, Ra¨ıs [24] pointed out that there is a gap in Westwick’s proof since the eigenvalues of B and C are assumed distinct. Poon gave the first elementary proof [23] of Westwick’s result.